Maths Blog

Tuesday, December 23, 2008

My friend was just telling me about this maths poem in the movie Harold and Kumar (2?) and I did some googling and found them! It's not too bad... But as with the previous pick up lines entry, try it at your own risk! :D

I fear that I will always beA lonely number like root three

The three is all that's good and right,Why must my three keep out of sightBeneath the vicious square root sign,I wish instead I were a nine

For nine could thwart this evil trick,with just some quick arithmetic

I know I'll never see the sun, as 1.7321Such is my reality, a sad irrationality

When hark! What is this I see,Another square root of a three

As quietly co-waltzing by,Together now we multiplyTo form a number we prefer,Rejoicing as an integer

Friday, December 19, 2008

The previous entry on prime numbers was too heavy for my liking! Typing it was fun but reading it was a chore! I shall edit my style of writing to put things across nice and short. Pardon me for the previous entry!

Saturday, December 13, 2008

What better way to start this intellectual pursuit of Mathematics than to start with something that all of us know of - Prime Numbers.

In school, we learn that prime numbers are numbers which are only divisible by 1 and itself. Hence, the prime numbers are 2, 3, 5, 7, 11, ... the list goes on and on. You might wonder why 1 isn't a prime number since according to the definition, 1 should be! However, many mathematicians dismiss 1 as a prime number and start with 2 in the series of prime numbers as it enables theorems to be elegantly stated. For example, prime factorization of 24 is uniquely set as 2^3 X 3^1. If we allow 1 into our list of prime numbers, prime factorization of 24 could be 1 X 2^3 X 3^1 or even 1^234567876543234567 X 2^3 X 3^1! Moreover, the definition of prime numbers states that they have to be divisible by 1 and itself. Maybe we should further add in a side note that 1 not equals to itself. Mathematicians have weird logical reasonings sometimes!

Moving on, prime numbers are very much to Mathematics what atoms are to Chemistry. Like atoms, which are building blocks of matter, all numbers can be expressed in the form of prime factorization as shown above. There is a name for this too! It's called the 'prime-number decomposition theorem'. This theorem states that every whole number greater than 1 can be written by multiplying prime numbers in exactly one way.

Having said so much, you might like to know that the greatest prime number as of 28/09/2008 is 2^43112609 - 1! That is close to 13 million digits long! In fact, the list of prime numbers goes on and on to infinity as proven by Euclid. In his book Elements (Book 9, Proposition 20), he stated that 'Prime numbers are more than any assigned multitude of prime numbers' and proceeded to write his proof down. Though quite short, I shall not bore you with sleep inducing proofs like this. If you're interested, you can read the footnote at the bottom.

The number 666, commonly known as the 'number of the beast' in the biblical book of Revelations, has some unexpected properties regarding prime numbers. It is the sum of the squares of the first 7 primes: 666 = 2^2 + 3^2 + 5^2 + 11^2 +13^2 + 17^2. Impressed? You might want to know that 666 is also the sum of palindromic cubes too: 666 = 1^3 +2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 5^3 + 4^3 + 3^3 + 2^3 + 1^3. And note: the keystone 6^3 in the centre is shorthand for 6 X 6 X 6! Need I say more?

Well. The amount of knowledge of prime numbers that is known is already so much, but there are still many unknown areas about it waiting to be explored. A famous one would be the 'Goldbach Conjecture'. Christian Goldbach conjectured that 'Every even number greater than 2 is the sum of two prime numbers.' For instance, 42 is an even number and is the sum of 5 and 37, both prime numbers. It can even be 11 + 31, 13 + 29 or 19 + 23. The conjecture is true for many numbers, but it has never been proved in general. Progress has been made towards the proof, and many mathematicians feel that the proof isn't far off. Chinese mathematician Chen Jingrun made a great step by proving his theorem that 'Every sufficiently large even number can be written as the sum of two primes of the sum of a prime and a semi-prime (a number which is the multiplication of two primes).' I hope I get to read the proof one day - not in chinese though because my chinese is horrible!

Ending this long entry about prime numbers, prime numbers can be broken down too, much like atoms into smaller units like quarks. In the realm of complex numbers, prime numbers like 5 can be expressed as a product of two numbers. For example, 5 = (1 - 2i) X (1 + 2i), where i is the square root of -1 of the imaginary number system. As a product of two Gaussian integers, prime numbers are not as unbreakable as was once supposed!

I hope I enlightened you somewhat on this entry about prime numbers and sparked some interest in you wanting to know more about the wonders of Mathematics. Though intrinsically basic, prime numbers offers stiff challenges for those who delve deeper into its roots. Feel free to comment on this entry. Cheers!

Footnote. Euclid's Elements (Book 9, Proposition 20)

Suppose that P is the largest prime, and consider the number N = (2 X 3 X 5 X ... X P) + 1. Either N is prime or it is not. If N is prime, we have produced a prime greater than P, which is a contradiction to our supposition. If N is not a prime, it must be divisible by some prime, say p, which is one of 2, 3, 5, ..., P. This means that p divides N - (2 X 3 X 5 X ... X P). But this number is equals to 1 and so p divides 1. This cannot be since all primes are greater than 1. Thus, whatever the nature of N, we arrive at a contradiction. Our original assumption of there being a largest prime P is therefore false. Conclusion: the number of prime is limitless.

Disclaimer: This entry is written based on the knowledge of the book '50 mathematical ideas you really need to know' by Tony Crilly.

Hello everyone! Welcome to the World of Mathematics! This is the first entry of this special project of mine that I have always wanted to start, something that is the first of its kind on the internet. When I was still schooling in secondary school and in junior college, I always wanted a place where I can get everything I need to know about my favourite subject, Mathematics, and more. A place where I can get Mathematics papers for free (always believed in sharing instead of buying them!) and also interesting Mathematics ideas, concepts and maybe even the history of Mathematics and Mathematicians that we do not learn in the textbooks. A place where there is someone readily available to help me with questions that I am stuck in and clarify my doubts about several Mathematics concepts. A few months back, a good friend of mine gave me a book for my birthday titled "50 Mathematical Ideas You Really Need To Know" by Professor Tony Crilly. I was so engrossed in the book which gave me several insights into this wonderful world of Mathematics that I want to share it with everyone out there who are interested. I went back to the idea I thought of in school days and after much deliberation, mathsblogsg was born.

I will trawl through the internet to bring you interesting Mathematics stuffs every day if possible. In mathsblogsg, there will be comics, jokes, maths posers, introduction to famous mathematicians and concepts you do not learn in school, history of maths, recent maths news and even sale of maths tees designed by my friend and I and many many more!

I have also set up a gmail account to cater to students who have Mathematics questions that they cannot solve or any Mathematics concepts they cannot understand. Simply drop an email to askmeamathsquestion@gmail.com and I will try and reply within 48 hours from the time the email is sent!

I will also be collating Mathematics papers from all the schools in Singapore too! If you are a student currently schooling in any secondary school or junior college in Singapore and wishes to share your school's examination papers with the rest of the people out there for free, scan in a copy of it (with answers I hope!) and send it to askmeamathsquestion@gmail.com. I will list down the papers available on the column on the right once I have received it and make it available to everyone! If you want a particular paper, simply drop an email to the said address to enquire about it and I will try to send it to you within 48 hours of receiving it.

I hope that you, the reader, will join me in spreading the love and wonders of Mathematics to everyone out there! Cheers!

About Me

I am a 20 year old chinese guy who aspires to teach Mathematics and wishes to spread the wonders of Mathematics to everyone!

I will trawl through the internet to bring you interesting Mathematics stuffs every week. In mathsblogsg, there will be comics, jokes, maths posers, famous mathematicians, history of maths, maths in everyday life, recent maths news and even sale of maths tees designed by me and my friend and many many more! Subcribe to mathsblogsg now or visit here every few days to get your dosage of Mathematics.

So sit back, relax, and let your clicking explore the wonderful world of Mathematics! Cheers! (:

Ask Me A Maths Question!

Ran into some trouble in your maths homework? Having difficulty understanding a particular maths concept that you have been staring at for hours to no avail? Ask Me A Maths Question now and I will get back to you in 1-2 days with (hopefully!) the solution to your problem or the best explanation to the concept you are having difficulty with!

Email me at askmeamathsquestion@gmail.com now!

Maths Tuition

If you're interested in getting me to tutor you mathematics, contact me at askmeamathsquestion@gmail.com to enquire about my availability. (:

Maths Resources

I don't believe in paying high prices just to get top schools examination papers so I'm providing this public good to everyone out there!

Currently, I have all the JC's Mathematics and Chemistry Prelims 2008 papers with me, together with my personal notes and formula lists for JC level.

If you wish to share your school's examination papers with the rest of the people out there, simply scan a copy of it (preferably with answers) and send it to me at askmeamathsquestion@gmail.com. Needless to say, if you want a copy of a particular school's papers, please feel free to enquire through the same email too!